invertiforms (version 0.1.2)

NormalizedLaplacian: Construct and use the Normalized Laplacian

Description

A convenience function to create NormalizedLaplacian S4 objects, which are useful for finding the normalized Laplacian of the adjacency matrix of a graph.

Usage

NormalizedLaplacian(A)
# S4 method for NormalizedLaplacian,sparseMatrix
transform(iform, A)
# S4 method for NormalizedLaplacian,sparseMatrix
inverse_transform(iform, A)

Value

• NormalizedLaplacian() creates a NormalizedLaplacian object.

• transform() returns the transformed matrix, typically as a Matrix::Matrix.

• inverse_transform() returns the inverse transformed matrix, typically as a Matrix::Matrix.

Arguments

A

A matrix to transform.

iform

An Invertiform object describing the transformation.

Details

We define the normalized Laplacian \(L(A)\) of an \(n \times n\) graph adjacency matrix \(A\) as

$$ L(A)_{ij} = \frac{A_{ij}}{\sqrt{d^{out}_i} \sqrt{d^{in}_j}} $$

where

$$ d^{out}_i = \sum_{j=1}^n \| A_{ij} \| $$

and

$$ d^{in}_j = \sum_{i=1}^n \| A_{ij} \|. $$

When \(A_{ij}\) denotes the present of an edge from node \(i\) to node \(j\), which is fairly standard notation, \(d^{out}_i\) denotes the (absolute) out-degree of node \(i\) and \(d^{in}_j\) denotes the (absolute) in-degree of node \(j\).

Note that this documentation renders most clearly at https://rohelab.github.io/invertiforms/.

Examples

library(igraph)
library(igraphdata)

data("karate", package = "igraphdata")

A <- get.adjacency(karate)

iform <- NormalizedLaplacian(A)

L <- transform(iform, A)
A_recovered <- inverse_transform(iform, L)

all.equal(A, A_recovered)